Define $D^{\text{#}}_{\bullet}(Mod(\mathcal{D}_X))$ to be the full subcategory of the derived category $D^{\text{#}}(Mod(\mathcal{D}_X))$ of complexes of $\mathcal{D}_X$-modules whose cohomology groups belong to $Mod_{\bullet}(\mathcal{D}_X)$, # = $+,-,b$, $X$ smooth algebraic variety over $\mathbb{C}$.

Is it true that $D^{\text{#}}_{\bullet}(Mod(\mathcal{D}_X))$ is equivalent to the category $D^{\text{#}}(Mod_{\bullet}(\mathcal{D}_X))$?

It is true that $Mod_{\bullet}(\mathcal{D}_X)$ are closed under kernels, cokernels and extensions because we are working over a Noetherian scheme. However, Kashiwara-Schapira Category and Sheaves states the equivalence under one more hypothesis. Apart from the answer, if you can quote any reference I would be glad.

EDIT: I forgot to say that $\bullet =$ quasi coherent modules or coherent modules (a quasi coherent $\mathcal{D}_X$ module is a $\mathcal{D}_X$ modules which is quasi coherent over $\mathcal{O}_X$, instead coherence is over $\mathcal{D}_X$)

$\begingroup$IMHO, the answer is UNKNOWN. If there is a good reason for it to be YES, it is known thanks to Exercise on p.153 of Gelfand-Manin or 2.42 and 3.4 of Huybrechts...$\endgroup$
– Bugs BunnyMar 29 '18 at 11:16

1 Answer
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I post it as an answer because I found it on a book: Theorem 1.5.7 of D-Modules, perverse sheaves, and representation theory by Hotta, Takeuchi, and Tanisaki states that the natural functors give equivalence or categories